Sliding surface searching method for slopes containing a potential weak structural surface

Aijun Yo,Zhizhou Tin,Yongjun Jin

aKey Laboratory of Urban Security and Disaster Engineering,Ministry of Education,Beijing University of Technology,Beijing 100124,China

bNorth China Power Engineering Co.,Ltd.,China Power Engineering Consulting Group,Beijing 100120,China

Sliding surface searching method for slopes containing a potential weak structural surface

Aijun Yaoa,*,Zhizhou Tiana,Yongjun Jinb

aKey Laboratory of Urban Security and Disaster Engineering,Ministry of Education,Beijing University of Technology,Beijing 100124,China

bNorth China Power Engineering Co.,Ltd.,China Power Engineering Consulting Group,Beijing 100120,China

A R T I C L E I N F O

Article history:

Received 5 March 2014

Received in revised form

10 March 2014

Accepted 16 March 2014

Available online 16 April 2014

Weak structural surface

Weak structural surface is one of the key factors controlling the stability of slopes.The stability of rock slopes is in general concerned with set of discontinuities.However,in soft rocks,failure can occur along surfaces approaching to a circular failure surface.To better understand the position of potential sliding surface,a new method called simplex-f i nite stochastic tracking method is proposed.This method basically divides sliding surface into two parts:one is described by smooth curve obtained by random searching,the other one is polyline formed by the weak structural surface.Single or multiple sliding surfaces can be considered,and consequently several types of combined sliding surfaces can be simulated.The paper will adopt the arc-polyline to simulate potential sliding surface and analyze the searching process of sliding surface.Accordingly,software for slope stability analysis using this method was developed and applied in real cases.The results show that,using simplex-f i nite stochastic tracking method,it is possible to locate the position of a potential sliding surface in the slope.

?2014 Institute of Rock and Soil Mechanics,Chinese Academy of Sciences.Production and hosting by Elsevier B.V.All rights reserved.

1.Introduction

Slope stability analysis is a very important issue for geotechnical engineers,and it has attracted extensive attention across the world (Lu et al.,2002;Wyllie and Mah,2004).The stability of rock slopes is in general concerned with set of discontinuities.However,in soft rocks failure can occur along surfaces approaching to a circular failure surface.Basically slope stability analysis should be conducted in two steps.The f i rst one is to f i nd out the position of the slope potential sliding surface,and the second is to analyze the slope stability in these surfaces.Limit equilibrium methods are very common method and widely applied to slope stability analysis with various slope shapes and engineering geological conditions(Ni, 2004;Shi and Luan,2013).However,numerical analysis methods are nowadays used predominantly in rock slopes to safety analysis even without pre-de f i ning slide planes.For soft rocks,the key issue is how to search the position of potential sliding surfaces.Once position of sliding surface is determined,satisfactory results can be obtained(Zhao,2006).

Many studies have been conducted on the sliding surface searching technology since the 1970s(Fang et al.,2007;Yao and Xue,2008).Regarding searching technology various kinds of searching methods were developed,including variation method, pattern searching algorithm,mathematical programming approach,dynamic programming method,random searching algorithm,arti f i cial intelligence method,etc.These methods make it possible to search a single sliding surface,and fruitful achievements are obtained.When weak structural planes or surfaces are present in the slope,a potential sliding surface usually consists of two parts: one is expressed by smooth curve obtained by random searching, and the other one can be a polyline formed by weak structural planes(Yin et al.,2007;Guo et al.,2013).In this way,the abovementioned methods are not suitable for searching the potential sliding surface,and a new method called simplex-f i nite stochastic tracking method is employed in the context.Engineering practice shows that the proposed method permits to solve the problem effectively.

2.Characteristics of slope containing weak structural plane

Weak structural planes or surfaces refer to the geological structure that controls the geometry and the position of slope(Li et al.,1996;Wang et al.,2006;Ren et al.,2008).It plays a crucial role in the stability evaluation of slopes(Zhang et al.,2001,2012).Inpractical projects,for the geometry of the model is usually adopted a straight line or a polyline to simulate the weak structural surface. A weak structural surface has the following characteristics:

(1)The origins of weak structural surfaces are very complex(Qian et al.,2006;Du,2013).Some weak structural surfaces are formed in the digenesis stage as shown in Fig.1a.Some are formed under the action of tectonic stress,such as the existence of discontinuities or faults within the rock slope,as shown in Fig.1b and c.Others are formed under the action of external forces(weathering,unloading,groundwater,blasting,etc.),just like cracks and mudded intercalation,as shown in Fig.1d.

(2)The shear strength of a weak structural surface is signi f i cantly low,induced by the presence of a certain amount of f i lling materials in the weak surface(Xie et al.,2006).

(3)As the strength of weak surface is signi f i cantly low,once the slope fails,it will slip along the potential sliding surface easily (Zhu et al.,2010).

3.Method for searching potential sliding surface

The basic principle ofsimplex-f i nite stochastic tracking method is to search and to optimize a potential sliding surface within the speci f i ed bound.In general,sliding body slips along the weak structural surface when the weak structural surface is observed in the toe or top of the slope.This paper focuses on the searching process ofthe slope containing pre f i xed weak structuralplane by means ofa simplex-f i nite stochastic tracking method.The combined sliding surface consists of two parts:circular sliding surface and fold-line plane(Liu et al.,1998).Its objective function can be de f i ned as

wherey=CombinSlip（x）is the expression of the combined sliding surface.

3.1.Mathematical model

The slope discretization model is shown in Fig.2,wherey=Slope（x）is the formula of the expression of slope lines.It was considered that the slope is divided into six data nodes,i.e.[Ni,i=1, 2,…,6].Each data node is described by the two-dimensional coordinates,(xi,yi).In addition,each node data must satisfy the following conditions:xi+1＞xiandyi+1≥yi.They=Wstrs（x）shown in Fig.2 is the expression of weak structural plane data, which are divided into three data nodes,[A,B,C],and each node data must satisfy the following condition:xC＞xB＞xA.In fact,the number of slope lines data and surface are determined by sitespeci f i c investigation.

The ground model can be discretized as shown in Fig.2 for two strata:stratum I is divided into four nodes[N5,N4,N7,N6],and stratum II is divided into six nodes[N3,N2,N1,N9,N8,N7].

Fig.1.The sketch of potential sliding surface.

Fig.2.The discretization model of slope.

3.2.Searching process of potential sliding surface

If a weak plane exists in slope,the sliding basically slips along the potential sliding surface,so the end pointEof the circular sliding surface would be located in the weak structural plane,as shown in Fig.2.

The f i rst step is to identify the starting pointSof the circular sliding surface,as shown in Fig.3.xS∈（x7，x8）and its expression is shown as

whereR1is a uniformly distributed random number,andR1∈(0,1). The second step is to identify the searching scope of the end pointEin the circular sliding surface,as shown in Fig.3.Its expression iswhereR2is a uniformly distributed random number,andR2∈(0,1).

Fig.3.The searching constraints diagram for potential sliding surface.

The third step is to identify the mid-pointTin the circular sliding surface;it must be on the line connecting the pointsOandMand satisfying the following conditions:

(1)The sliding surface should not be convex,i.e.xT≥xM.

(2)The sliding surface and the slope lines should not have intersection point,just like the left dotted line shown in Fig.3.

(3)The slope of tangent line for the start pointSin the circular sliding surface should not be negative,and not less than that of the f i rst section of the weak structural plane.

We can obtainxTminby conditions(1)and(2),andxTmaxby condition(3).The expression of mid-pointTis

whereR3is the uniformly distributed random number,andR3∈(0, 1);KJis the slope of the line connecting pointsOandM.

The fourth step is to construct a circular sliding surface.According to the above three steps,we have obtained pointsS,T,E. The determinant circular sliding surface is written as

The f i fth step is to judge the effectiveness of circular sliding surface.The circular sliding surface obtained by random searching, as a part of the combined sliding surface,is ineffective when it intersects weak structural plane and the nodeCof weak structural plane falls inside of the circle.It is easily sliding along the weak structural surface,because the strength of weak surface is signi f icantly lower than surrounding rock and soil.Just like the circular sliding surface(arcSTE2E1E)is ineffective,but part of the circular sliding surface(arcSTE2)is effective.So the combined sliding surface consists of arcSTE2,polylinesE2BandBAas shown in Fig.4.

The circular sliding surface,as a part of the combined sliding surface,is also ineffective when it intersects with weak structural plane and the nodeCof weak structural plane falls inside of the circle as shown in Fig.5.The slope sliding along theSE1Eis impossible.But we can get arcSTCinstead of arcSE1E.ArcSTCobtained by random search,the pointCis the end point of the weak structural plane,the pointSis obtained the last random search.So the combined sliding surface consists of arcSTC,polylinesCBandBAas shown in Fig.5,whereTis the point obtained by searching.

Fig.4.The f i rst kind of ineffective circular sliding surface.

Fig.5.The second kind of ineffective circular sliding surface.

The sixth step is to calculate the safety factor of slope:calculate the approximation of the safety factor of slope on the basis of objective function,and then optimize it,and f i nally determine the position of sliding surface and its safety factor.

4.Analysis of an engineering example

To better understand the in f l uence of weak surfaces on slope stability,the developed software DL-SLOPE V1.0 was used to analyze the stability of a slope in Fenshui town,Wanzhou district, Chongqing,China.The software uses the transfer coef f i cient method,which is also called imbalance thrust force method or polygon method(Pan,1980),and the simpli f i ed algorithms show off its simple calculation,applicability and convenience to good effect,besides,it has been very popular in the irrigation department and the railway department of China.

4.1.Brief description of the project

At the project site,there is a reservoir in the northeastern side, and a national highway along the north side.The site belongs to hilly area and has topographic relief,which is caused by early tectonic denudation.Besides,this project site is surrounded by various extremely important buildings and structures.Therefore,if the slope fails,it can cause heavy casualties and great economic losses.

The elevation of the project site is about 654.3 m.The formation fromtop to bottom can be divided into following four layers and the geo-mechanical parameters of all layers are listed in Table 1:

(1)The f i rst layer mainly consists of silty clay,which is brown, brown yellow or gray-yellow in color.It has a wide distribution and its shear strength is low.

(2)The second layer mainly consists of silt,which is gray-brown or gray-yellow in color,and the average thickness is about 0.5 m. Its shear strength is much lower than others,which can be de f i ned as the weak structural plane.

(3)The third layer mainly consists of weathered sandstone,which is off-white or gunmetal-gray in color,and the averagethickness is about 11.0 m.This layer has a wide distribution,and clayey mineral is dominant but with high shear strength.

Table 1Geo-mechanical parameters of the four layers.

(4)The fourth layer mainly consists of slightly weathered sandstone,which is off-white in color,and the average thickness is about 12.5 m.Feldspar is dominant in this layer,and then is quartz,so the shear strength of the fourth layer is higher than that of the third layer.

Fig.6.The geometric model of the most unfavorable slope.

Table 2The design parameters of anchor cables.

Fig.7.The results for two cases:(a)considering weak surface;(b)without considering weak surface.

The project site is located in an area of seismic intensity VI. According to theTechnical Code for Building Slope Engineering(GB50330-2002)(MOHURD,2002),it is not necessary to conduct seismic calculations.

4.2.Stability analysis

In view of the importance of the surrounding buildings,the slope must be maintained in a stable condition.According to MOHURD(2002),the design safety class of the slope is I,and in consequence the safety factor of the slope should be not less than 1.30.The geometric model for the most unfavorable slope is shown in Fig.6.

Reinforcement techniques using anchor cables were adopted in order to improve the safety factor of the slope.The design parameters of anchor cables are listed in Table 2.

The safety factors of slope were calculated respectively for two cases taking or not into account the in f l uence of weak surface.The results are shown in Fig.7.

When considering the in f l uence of weak surface,the position of the potential sliding surface is the combined sliding surface and the safety factor of slope is 1.302.Without considering the in f l uence of weak structural plane,the position of the potential sliding surface is the circular sliding surface and the safety factor of slope is 1.721.

5.Conclusions

From the situations presented in the paper,the following conclusions can be drawn:

(1)The weak structural surface is directly associated with the position of the potential sliding surface due to its low shear strength.In this regard,it should be considered when analyzing the slope stability.

(2)The simplex-f i nite stochastic tracking method is simple and can be coded conveniently.The potential sliding surface searched by the proposed method can be used to re f l ect the real engineering geological conditions.The case study demonstrates that considering the in f l uence of weak surface when using the simplex-f i nite stochastic tracking method is very important in the slope stability analysis.

Con f l ict of interest

We wish to con fi rm that there are no known con fl icts of interest associated with this publication and there has been no signi fi cant fi nancial support for this work that could have in fl uenced its outcome.

Acknowledgment

The authors are grateful for f i nancial support from the National Natural Science Foundation of China under Grant No.50978007.

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Dr.Aijun Yaois working as a professor at Beijing University of Technology(BJUT),China.His research interests cover slope engineering,foundation pit engineering,geology disaster evaluation,detection technique,urban environmental geotechnology,etc.Contact information:No.100,Pingleyuan,Chaoyang District,Beijing,100124, China.Tel.:+86 13683269395.E-mail:yaj@bjut.edu.cn.

*Corresponding author.Tel.:+86 13683269395.

E-mail address:yaj@bjut.edu.cn(A.Yao).

Peer review under responsibility of Institute of Rock and Soil Mechanics,Chinese Academy of Sciences.

Potential sliding surface

Slope stability

Simplex-f i nite stochastic tracking method